What is Metric: Definition and 1000 Discussions

METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model developed by the University of Idaho, that uses Landsat satellite data to compute and map evapotranspiration (ET). METRIC calculates ET as a residual of the surface energy balance, where ET is estimated by keeping account of total net short wave and long wave radiation at the vegetation or soil surface, the amount of heat conducted into soil, and the amount of heat convected into the air above the surface. The difference in these three terms represents the amount of energy absorbed during the conversion of liquid water to vapor, which is ET. METRIC expresses near-surface temperature gradients used in heat convection as indexed functions of radiometric surface temperature, thereby eliminating the need for absolutely accurate surface temperature and the need for air-temperature measurements.

The surface energy balance is internally calibrated using ground-based reference ET that is based on local weather or gridded weather data sets to reduce computational biases inherent to remote sensing-based energy balance. Slope and aspect functions and temperature lapsing are used for application to mountainous terrain. METRIC algorithms are designed for relatively routine application by trained engineers and other technical professionals who possess a familiarity with energy balance and basic radiation physics. The primary inputs for the model are short-wave and long-wave thermal images from a satellite e.g., Landsat and MODIS, a digital elevation model, and ground-based weather data measured within or near the area of interest. ET “maps” i.e., images via METRIC provide the means to quantify ET on a field-by-field basis in terms of both the rate and spatial distribution. The use of surface energy balance can detect reduced ET caused by water shortage.
In the decade since Idaho introduced METRIC, it has been adopted for use in Montana, California, New Mexico, Utah, Wyoming, Texas, Nebraska, Colorado, Nevada, and Oregon. The mapping method has enabled these states to negotiate Native American water rights; assess agriculture to urban water transfers; manage aquifer depletion, monitor water right compliance; and protect endangered species.

View More On Wikipedia.org
Recent contents Top users
  1. S

    Why is Scalar Cam Built with Second-Order Derivative of Metric Ricci Scalar?

    hi why only scalar cam build with second order of derivative of metric is Ricci scalar? thanks
  2. alyafey22

    MHB Continuous mapping of compact metric spaces

    Let $f$ be a continuous mapping of a compact metric space $X$ into a metric space $Y$ then $f$ is uniformly continuous on $X$. I have seen a proof in the Rudin's book but I don't quite get it , can anybody establish another proof but with more details ?
  3. Philosophaie

    Finding the 4x4 Cofactor of a Covariant Metric Tensor g_{ik}

    If I have a 4x4 Covarient Metric Tensor g_{ik}. I can find the determinant: G = det(g_{ik}) How do I find the 4x4 Cofactor of g_ik? G^{ik} then g^{ik}=G^{ik}/G
  4. H

    Complete countable metric space

    Homework Statement It is clear that a countable complete metric space must have an isolated point,moreover,the set of isolated points is dense.what example is there of a countable complete metric space with points that are not isolated? Homework Equations The Attempt at a Solution
  5. M

    Is Schwarzschild metric more intuitive?

    The textbooks claim that the weak field (Newtonian) metric is more intuitive than the Schwarzschild metric, but I don’t agree.The time correction factor for the weak field metric is the same as that for the Schwarzschild metric. But for the length correction factor for the weak field metric is...
  6. Mandelbroth

    Metric Tensor on a Mobius Strip?

    I was bored, so I tried to do something to occupy myself. I started going through withdrawal, so I finally just gave in and tried to do some math. Three months of no school is going to be painful. I think I have problems. MATH problems. :-p Atrocious comedy aside, Spivak provides a parametric...
  7. R

    Calculate Christoffel Symbols of 2D Metric

    Homework Statement Consider metric ds2 = dx2 + x3 dy2 for 2D space. Calculate all non-zero christoffel symbols of metric. Homework Equations \Gammajik = \partialei / \partial xk \times ej The Attempt at a Solution Christoffel symbols, by definition, takes the partial of each...
  8. I

    Metric on torus induced by identification of points on plane

    Hi all, Perhaps I'm asking the wrong question but I am wondering about the relationship between different definitions of, for the sake of argument, the torus. We can define it parametrically (or as a single constraint) and from there work out the induced metric as with any surface. But...
  9. F

    Lorentz invariance of an equation (metric)

  10. G

    Proving a function is bounded and continuous in a metric space.

    Homework Statement Let (X,d) be any metric space. Fix a in X and for each x in X define fx:X→ℝ by: fx(z)=d(z,x)-d(z,a) for all z in X. Show that fx(z) is bounded and continuous. The Attempt at a Solution I can't figure out how to tell if it is bounded. Any hints? I'm sure...
  11. E

    Wrong signs Ricci tensor components RW metric tric

    Hi, I am working through GR by myself and decided to derive the Friedmann equations from the RW metric w. ( +,-,-,-) signature. I succeeded except that I get right value but the opposite sign for each of the Ricci tensor components and the Ricci scalar e.g. For R00 I get +3R../R not -3R../R . I...
  12. B

    Trying to Prove that Given Function is a Metric

    Hello! I'm trying to prove that ##d_1\left({x,y}\right)=\max\{\left|{x_j-y_j}\right|:j=1,2,...,k\}## is a metric. I know that since ##d_1\left({x,y}\right) = \left|{x_j-y_j}\right|## for some ##j## that ##d_1\left({x,y}\right) \geq 0## and since ##\left|{x_j-y_j}\right| =...
  13. 4

    Question about value of the metric tensor and field strength

    Is it the value of the metric tensor that determines the strength of a gravitational field at a specific point in spacetime?
  14. P

    MHB How Do You Differentiate the Modulus of a Complex Number in Riemannian Metrics?

    I'm interested in part iv) on the attachment. This is my work so far: e=(1,0) and e'=(0,1) form a basis of the tangent space at any point z=(x,y). Making the identification (x,y)->x+iy, we get g(e,e')=0 and g(e,e)=g(e',e')=$\frac{1}{im(z)^2}$. a(t)=z+t and b(t)=z+it are generating curves for...
  15. A

    Anisotropic instead of isotropic metric deriving acceleration

    In this documentation from Nasa a procedure to get to what I guess is the gravitational acceleration according to the post-Newtonian expansion at the 1PN-level for the spherically symmetric case is found: http://descanso.jpl.nasa.gov/Monograph/series2/Descanso2_all.pdf The procedure is...
  16. P

    Proving a sequence is a cauchy sequence in for the 7 -adic metric

    Homework Statement Show that the sequence (xn)n\inN \inZ given by xn = Ʃ from k=0 to n (7n) for all n \in N is a cauchy sequence for the 7 adic metric. Homework Equations In a metric space (X,dx) a sequence (xn)n\inN in X is a cauchy sequence if for all ε> 0 there exists some M\inN such...
  17. shounakbhatta

    Understanding the Minkowski Metric: Explained Step-by-Step for Beginners

    Hello, I don't know whether I have mentioned the subject line properly. Many times while reading over General Relativity I come across the following equation: ds^2=dx1^2+dx2^2+dx3^2+dx4^2 =dx^2+dy^2+dz^2-c^2dt^2. Now, my question from the above equation is: (a) Are we putting...
  18. M

    He most general form of the metric for a homogeneous, isotropic and st

    What is the most general form of the metric for a homogeneous, isotropic and static space-time? For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) signature) ds^2=dt^2+a^2(t)g_{ij}(\vec x)dx^idx^j Now the static condition. If I'm not mistaken...
  19. M

    Second derivative of a metric and the Riemann curvature tensor

    I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial coordinates) R_{abcd}=\frac{1}{2} (g_{ad,bc}+g_{bc,ad}-g_{bd,ac}-g_{ac,bd}) where ",_i"...
  20. G

    Confusion about continuity in metric spaces

    I'm confused about the the definition of a function not being continuous. Is it correct to say f(x) is not continuous at x in the metric space (X,d) if \existsε>0 such that \forall\delta there exists a y in X such that d(x,y)<\delta implies d(f(x),f(y))>ε Is y dependant on \delta? It...
  21. C

    Solving field equations and the nonphysical nature of the metric. (GR)

    There seems to be an emphasis in several books on general relativity that the metric (components) in itself does not reflect anything physical, only our choice of coordinates. On the other hand it can seem like the authors, instead of being true to this, treat the metric (components) as...
  22. M

    Given a metric space (X,d), the set X is open in X. HELP

    I must be overlooking something! Given a metric space (E,d), the improper subset E is open in E. How? Here is my understanding: 1) We call a set S(subset of E) open iff for all x(element of S) there exist epsilon such that an open ball of radi epsilon centered about s is wholly contained in...
  23. WannabeNewton

    Wald Problem 6.3: Reissner-Nordstrom Metric

    Hi guys. This question is related to Problem 6.3 in Wald which involves deriving the Reissner-Nordstrom (RN) metric. We start with the source free Maxwell's equations ##\nabla^{a}F_{ab} = 0,\nabla_{[a}F_{bc]} = 0## in a static spherically symmetric space-time which, in the coordinates adapted to...
  24. S

    Rindler metric and proper acceleration.

    Homework Statement Show that observers whose world-lines are the curves along which only ξ varies undergo constant proper acceleration with invariant magnitude ημν(duμ/dτ )(duν/dτ ) = 1/χ2. Homework Equations ds2 = −a2χ2dξ2 + dχ2 + dy2 + dz2 xμ = {ξ, χ, y, z} t = χsinh(aξ) x...
  25. T

    Schwarzschild Metric & Satellite Orbits: A Question

    Thanks in advance - this problem has been bothering me for a while! I'm working with an unpowered spaceship orbiting a large mass M. The orbit is circular and it is following the geodesic freely. It has an orbit radies of r = R. My question is this. The metric of the space-time curvature...
  26. S

    Raising and Lowering Indices of the Metric

    Homework Statement "Evaluate: g^{\mu \nu} g_{\nu \rho} where ds^2 = g_{\mu \nu} dx^\mu dx^\nu , ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 " Homework Equations None necessary, just a notation issue The Attempt at a Solution Just using raising and lowering rules, I imagine each raising and...
  27. D

    Why is Motion Through Space Limited in Relation to Motion Through Time?

    Looking for clarification on something. Take the metric in this form ds^2 = dt^2 - dx^2 Brian Green likes to say a null path is one where motion through space is shared equally with motion through time, or (ds=0). Motion through space is not allowed that is "faster" than motion...
  28. S

    Demonstrating Metric Kerr Expression

    hi , how i can demonstrate the expression of metric kerr?
  29. B

    Gravitational time dilation from the metric

    How does one go about finding what the gravitational time dilation is from the metric? Is it simply t'/t_0=1/\sqrt{g_{tt}}? It seems that could be true for static metrics, but perhaps not more dynamic ones like the Kerr metric. My confusion on this arises on how to treat the time cross terms...
  30. Z

    Metric tensor - index manipulation

    hello, Do I have the right to perform the following : gjo,i + g0i,j = (gj0 δij + g0i ),j = (2 g0i ),j Thank you, Clear skies,
  31. G

    Subspaces and interiors of metric spaces problem.

    Homework Statement If S is a subspace of the metric space X prove (intxA)\capS\subsetints(A\capS) where A is an element of ΩX(Open subsets of X) The Attempt at a Solution So intxA=\bigcupBd(a,r) where d is the metric on X and the a's are elements of A and I think...
  32. G

    Does Average Linkage satisfy the properties of metric space?

    Homework Statement A dissimilarity measure d(x, y) for two data points x and y typically satisfy the following three properties: 1. d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y 2. d(x, y) = d(y, x) 3. d(x,z) ≤ d(x, y) + d(y,z)The following method has been proposed for measuring the...
  33. G

    Proving {x} is a closed set in a metric space

    Hi everyone, I posted this a couple days ago and didn't get a response, so I thought I'd try again. Let me know if something about this is confusing. Thanks! Homework Statement Let X be a metric space and let x\in{X} be any point. Prove that the set \left\{x\right\} is closed in X...
  34. MathematicalPhysicist

    Calculating the metric tensor.

    Suppose, I have the next metric: g = du^1 \otimes du^1 - du^2 \otimes du^2 And I want to calculate g(W,W), where for example W=\partial_1 + \partial_2 How would I calculate it? Thanks.
  35. N

    Metric constraints in choosing coordinates

    Hello all, I've been puzzled by this problem for some time now and was wondering if anyone here could help me out. Textbooks on GR (specifically when going into gravitational waves) tend not to elucidate this. It's often taken for granted that through the gauge diffeomorphism invariance (or...
  36. G

    Metric topolgy comprehension question

    Given the proposition: A subset U of a metric space is an open set iff U is a union of open balls. Then we are told that it follows from the above proposition that two metrics are equivalent if each open ball of either of the metric topolgies is an open set of the other topology. This is...
  37. L

    When do SO(2) actions on the circle in the plane determine a metric?

    a metric on the plane determines an action of SO(2) on is unit circle by rotation. Suppose one starts with a free transitive action of SO(2) on a circle. When does this come from a metric? Always?
  38. J

    How to manipulate the determinant of metric tensor?

    How to calculate something relating to the determinant of metric tensor? for example, its derivative ∂_{λ}g. and how to calculate1/g* ∂_{λ}g, which is from (3.33) in the book Spacetime and Geometry, in which the author says that it can be related to the Christoffel connection.
  39. R

    No metric on S^2 having curvature bounded above or below by 0

    So I ran into a question; Show that there is no metric on S^2 having curvature bounded above by 0 and no metric on surface of genus g which is bounded below by 0. honestly I have no idea what is going on here. I know that a Genus is the number of holes in some manifold or the number of...
  40. P

    Varying determinant of a metric

    Hi can anyone explain how to find \delta \sqrt{-g} when varying with respect to the metric tensor g^{\mu\nu}. i.e why is it equal to \delta \sqrt{-g} = -\frac{1}{2} \sqrt{-g}g_{\mu\nu} \delta g^{\mu \nu}
  41. M

    Can Metrics Reveal Flat Spaces Without Computing the Riemann Tensor?

    If you are given a metric gαβ and you are asked to find if it describes a flat space, is there any way to answer it without calculating the Riemman Tensor Rλμνσ? and how can I find for that given metric the coordinate transformation which brings it in conformal form? For example I'll give you...
  42. R

    Does This Sequence Converge in the 5-adic Metric?

    Metric Space and Topology HW help! Let X be a metric space and let (sn )n be a sequence whose terms are in X. We say that (sn )n converges to s \ni X if \forall \epsilon > 0 \exists N \forall n ≥ N : d(sn,s) < \epsilon For n ≥ 1, let jn = 2[(5^(n) - 5^(n-1))/4]. (Convince yourself...
  43. tsuwal

    Metric Spaces: Exercise 1.14 from Introduction to Topology (Dover)

    Homework Statement X is a metric and E is a subspace of X (E\subsetX) The boundary ∂E of a set E is defined to be the set of points adherent to both E and the complement of E, ∂E=\overline{E}\cap(\overline{X\E}) (ignore the red color, i can't get it out) Show that E is open if and only...
  44. H

    The illusory generality of metric spaces

    So I was flipping through Lang's Undergraduate Analysis and noticed the absence of the important concept of metric spaces. I checked the index and was referred to problem 2, Chapter 6 Section 2. There he defines a metric space, what a bounded metric is, and gives a few straightforward problems...
  45. B

    Comparing Norms & Metrics: Axioms & Differences

    Are the axioms of a Norm different from those of a Metric? For instance Wikipedia says: a NORM is a function p: V → R s.t. V is a Vector Space, with the following properties: For all a ∈ F and all u, v ∈ V, p(av) = |a| p(v), (positive homogeneity or positive scalability). p(u + v) ≤ p(u)...
  46. B

    Could someone specify a metric on (0,1)

    could someone specify a metric on (0,1) that defines (the same topology) as the abs. value (i.e. usual) metric and makes this open interval into a complete set? Thanks
  47. P

    Metric Properties of a Simple ds^2 Equation

    Am i correct in thinking that in a simple ds^2 =dt^2 - a^{2}d\underline{x}^{2} metric that: g_{\mu\nu}=diag(1,-a^2,-a^2,-a^2) and g^{\mu\nu}=diag(1,-\frac{1}{a^2},-\frac{1}{a^2},-\frac{1}{a^2}) thx
  48. G

    Closed and Open Subsets of a Metric Space

    Homework Statement Let X be an infinite set. For p\in X and q\in X, d(p,q)=1 for p\neq q and d(p,q)=0 for p=q Prove that this is a metric. Find all open subsets of X with this metric. Find all closed subsets of X with this metric. Homework Equations NA The Attempt at a...
  49. P

    Varying determinant of a metric

    Hi does anyone know how to calculate: \delta (det|g_{\mu\nu}|) or simply \delta g
  50. G

    Bounded Subsets of a Metric Space

    Homework Statement Let X be a metric space and let E be a subset of X. Show that E is bounded if and only if there exists M>0 s.t. for all p,q in E, we have d(p,q)<M. Homework Equations Use the definition of bounded which states that a subset E of a metric space X is bounded if there exists...
Back
Top